Question
Identify all of the transformations for the function f(x) = 3cos(−x + π3)−2
. Check all that apply
Responses
Vertical stretch by 3
Vertical stretch by 3 - correct
vertical shift up 2 units
vertical shift up 2 units - no response given
vertical shift down 2 units
vertical shift down 2 units - correct
reflection over the x-axis
reflection over the x-axis - incorrect
reflection over the y-axis
reflection over the y-axis - not selected, this is the correct answer
horizontal shift to the right π3
horizontal shift to the right pi over 3 - not selected, this is the correct answer
horizontal shift to the left π3
. Check all that apply
Responses
Vertical stretch by 3
Vertical stretch by 3 - correct
vertical shift up 2 units
vertical shift up 2 units - no response given
vertical shift down 2 units
vertical shift down 2 units - correct
reflection over the x-axis
reflection over the x-axis - incorrect
reflection over the y-axis
reflection over the y-axis - not selected, this is the correct answer
horizontal shift to the right π3
horizontal shift to the right pi over 3 - not selected, this is the correct answer
horizontal shift to the left π3
Answers
GPT-4o mini
Let's analyze the transformations for the function \( f(x) = 3\cos(-x + \frac{\pi}{3}) - 2 \).
1. **Vertical Stretch by 3**: The coefficient \( 3 \) in front of the cosine function indicates a vertical stretch by a factor of 3. This is correct.
2. **Vertical Shift Up 2 Units**: The term \(-2\) at the end of the function indicates a vertical shift down, not up. This statement is incorrect.
3. **Vertical Shift Down 2 Units**: The term \(-2\) indicates a vertical shift down by 2 units. This is correct.
4. **Reflection Over the X-Axis**: There is no negative sign in front of the whole function, only in front of the \( x \) in the cosine argument, so there is no reflection over the x-axis. This is incorrect.
5. **Reflection Over the Y-Axis**: The term \(-x\) in the cosine argument indicates a reflection over the y-axis. This is correct.
6. **Horizontal Shift to the Right \( \frac{\pi}{3} \)**: The term \( -x + \frac{\pi}{3} \) can be rewritten as \( -(x - \frac{\pi}{3}) \). Thus, this indicates a shift to the right by \( \frac{\pi}{3} \). This is correct.
7. **Horizontal Shift to the Left \( \frac{\pi}{3} \)**: Since we established that the function shifts to the right by \( \frac{\pi}{3} \), this is incorrect.
### Summary of Correct Answers:
- Vertical stretch by 3
- Vertical shift down 2 units
- Reflection over the y-axis
- Horizontal shift to the right \( \frac{\pi}{3} \)
The other responses regarding vertical shift up 2 units, reflection over the x-axis, and horizontal shift to the left \( \frac{\pi}{3} \) are incorrect.
1. **Vertical Stretch by 3**: The coefficient \( 3 \) in front of the cosine function indicates a vertical stretch by a factor of 3. This is correct.
2. **Vertical Shift Up 2 Units**: The term \(-2\) at the end of the function indicates a vertical shift down, not up. This statement is incorrect.
3. **Vertical Shift Down 2 Units**: The term \(-2\) indicates a vertical shift down by 2 units. This is correct.
4. **Reflection Over the X-Axis**: There is no negative sign in front of the whole function, only in front of the \( x \) in the cosine argument, so there is no reflection over the x-axis. This is incorrect.
5. **Reflection Over the Y-Axis**: The term \(-x\) in the cosine argument indicates a reflection over the y-axis. This is correct.
6. **Horizontal Shift to the Right \( \frac{\pi}{3} \)**: The term \( -x + \frac{\pi}{3} \) can be rewritten as \( -(x - \frac{\pi}{3}) \). Thus, this indicates a shift to the right by \( \frac{\pi}{3} \). This is correct.
7. **Horizontal Shift to the Left \( \frac{\pi}{3} \)**: Since we established that the function shifts to the right by \( \frac{\pi}{3} \), this is incorrect.
### Summary of Correct Answers:
- Vertical stretch by 3
- Vertical shift down 2 units
- Reflection over the y-axis
- Horizontal shift to the right \( \frac{\pi}{3} \)
The other responses regarding vertical shift up 2 units, reflection over the x-axis, and horizontal shift to the left \( \frac{\pi}{3} \) are incorrect.